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Questions tagged [algebraic-topology]

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

Algebraic topology is a mathematical subject that associates algebraic objects to topological spaces which are invariant under homeomorphism or homotopy equivalence. Often, these associations are functorial so that continuous maps induce morphisms between appropriate algebraic objects.

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21356 questions
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A covering space of "8" defines the commutator subgroup of F2

Let M be the squares space $(R\times Z) \cup (Z\times R)$ covering the 8 space (pointwise union of two circles) by calling one circle of the 8 (path starting and ending at the dot of intersection of the two circles) x and the other y, and declaring…
Idan
  • 782
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fundamental group of complement in the thickened torus

What is the fundamental group of $(T \times I ) \backslash J$ where $J$ is a closed loop going around the vertical $\mathbb{S}^{1}$ of the thickened torus twice.
jojd03ud
  • 21
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Does there exist a continuous function from the unit interval to a circle with "closure"?

Unit interval $I$ and $S^1$ have different topology - if one identifies the opposite ends of $I$ to make $S^1$, lots of points which were "far" in $I$ (in terms of a number of shared open sets), become "near" in $S^1$. Is such a function $f: I \to…
Entsy
  • 35
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Under what circumstances is $H^p(B,H^q(F,R))\cong H^p(B,R)\otimes H^q(F,R)$?

The above question comes from the Serre spectral sequence in cohomology. One would like to write something like $H^p(B,H^q(F,R))\cong Hom(H_p(B;\mathbb{Z});Hom(H_q(F;\mathbb{Z});R))\cong Hom(H_p(B;\mathbb{Z})\otimes H_q(F;\mathbb{Z});R)\cong…
Takirion
  • 1,520
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Attaching an n-cell to a singleton

Let $Y$ be a Hausdorff space, and let $f:S^{n-1} \to Y$ be continuous. Then $D^n \coprod_f Y$ is called the space obtained from $Y$ by attaching an n-cell (denoted $e^n$) via $f$ and is denoted $Y_f$ The charactersitc map $\Phi$ is the…
Juan S
  • 10,268
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Long exact sequence for compact manifolds with boundary

Consider the cotangent bundle $T^*S^2$ equipped with the Eguchi-Hanson metric. Since the metric is ALE and the cone at infinity is $\mathbb R^4/\mathbb Z_2$, we can define a compact manifold $D$ which is the contraction of $T^*S^2$ such that…
Totoro
  • 829
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Sources for some algebraic topology

I've been looking at some quals problems for algebraic topology that I found online. The problem is that I don't know if I can solve them with the amount of algebraic topology that I know, but nevertheless, they seem interesting. Also I know my…
dstt
  • 1,089
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how to prove $L(m)$ is a orientable 3-manifold and compute $H^{*}(L(m))$

Suppose $S^3=\{(z_1,z_2)\in \mathbb C^2\mid|z_1|^2+|z_2|^2=1\} $ there is a $\mathbb Z/m$ group action on $S^3$: $$\phi:\mathbb Z/m \times S^3 \rightarrow S^3:\phi(k,(z_1,z_2))=e^{\frac{2k\pi }{m}\dot i} (z_1,z_2)$$ Let $L(m)$ be the quotiont…
bruce lee
  • 51
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Surjective function from D2 to s1?

I have to (show or) disprove that there is no continuous surjection from $D^2$ (unit closed disk of $\mathbb{R}^2$) to $S^1$ (unit circle)? I would like to say that if $f: X \to Y$ is surjective, then $f_*: \pi_1(X) \to \pi_1(Y)$ is surjective too,…
Thomas
  • 1,124
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If $q\circ p$ is a normal covering map then $p$ also is a normal covering map.

Take $p:F\to E,q:E\to X$ maps such that $q$ and $q\circ p$ are covering maps. If $X$ is locally path-connected then we know that $p$ is a covering map. I want to prove that if $q\circ p$ is normal then $p$ is normal. For this, I took $e\in E$,…
LeviathanTheEsper
  • 3,163
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Question On Cech Cohomology

In section 16 of the topology book of Bott and Tu, there is a path fibration $\Omega S^2 \to PS^2 \to S^2$. The $E_2$ page of the spectral sequence of this fibration is $$E_2^{p,q}=H^p(S^2,H^q(\Omega S^2)).$$ This is a Cech Cohomology of $S^2$ with…
Proton
  • 1,095
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If $q\circ p$ and $q$ are covering maps then $p$ is also a covering map.

I was trying to do this exercise from Hatcher's book: Let $p:F\to E,q:E\to X$ maps such that $q$ and $q\circ p$ are covering maps. If $X$ is locally path-connected, then show that $p$ is a covering map. I tried it as follows: Let $e\in E$. I must…
LeviathanTheEsper
  • 3,163
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Let $X$ be the complex obtained by gluing two discs to $S^1$ via maps of relatively prime degree. $X$ is simply connected.

Let $X$ be the complex obtained by gluing two discs to $S^1$ via maps of relatively prime degree, say by $f:\partial D_1 \rightarrow S^1$ of degree $p$, and $g:\partial D_2 \rightarrow S^1$ of degree $q$. I would like to show that $X$ is simply…
Functor
  • 179
2
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2 answers

is there a continuous surjective map $f : T \rightarrow T$ such that the induced homomorphism $f^* : H1(T,x) \rightarrow H1(T,x)$is the zero-map.

Let $T = \mathbb{S}^1 \times \mathbb{S}^1$ be a torus and $x \in T$. Prove or disprove: There exists a continuous surjective map $f : T \rightarrow T$ such that the induced homomorphism $f^* : H_1(T,x) \rightarrow H_1(T,x)$is the zero-map. I have…
TheGeometer
  • 2,515
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Universal Covering of $S^1 \vee S^1 \vee S^2$

I've seen that the universal covering of $S^1 \vee S^1 \vee S^2$ is infinitely many $S^2$'s chain together by line segments where the endpoints are identified. But when we are wedge two copies of $S^1$, I'm not sure how to make the two line segments…
nekodesu
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